How to make transparent box with shadow?

To avoid seeing the shadow in the transparent area, you can put the button inside a knockout transparency group. If the background and the shadow are in the same knockout group, the group "knocks out" the shadow:(I added some background text to show the transparency of the button and text opacity to increase the contrast)

\documentclass{article}
\usepackage{kantlipsum}
\usepackage{tikz}
\usetikzlibrary{shadows}

\begin{document}
    \begin{tikzpicture}[baseline=(char.base)]
      \node at (0,0) {\fontsize{5pt}{0pt}\selectfont\parbox{4cm}{\kant[1]}};
      \begin{scope}[transparency group=knockout]
        \node(char)[draw,fill=white,rectangle, rounded corners = 2pt,drop shadow={opacity=.5, shadow xshift=0.1cm},% blur shadow = {shadow blur steps = 5}
        fill opacity = 0.4, text opacity = 0.8] {Shadow};
      \end{scope}[transparency group=knockout]
    \end{tikzpicture}
\end{document}

Warning: Knockout groups are not supported by a lot of PDF viewers. Especially Acrobat is the only viewer I know supporting this.

A tcolorbox with a beamer style is a handy option of a more realistic appearance of translucent glass with shadow:

\documentclass[12pt]{article}
\usepackage{lipsum}
\usepackage[most]{tcolorbox}
\tcbset{beamer,boxrule=1pt,tikz={opacity=0.8,transparency group},fontupper=\bfseries\sffamily}
\begin{document}
\lipsum[1]
\vspace{-12em}\centering
\tcbox{Shadow}
\vspace{3em}
\tcbox[shadow={1mm}{-1mm}{-.5mm}%
{fill=cyan!75,opacity=0.15}]{Shadow}
\end{document}